HOMEWORK, 30 SEPTEMBER 2013
1. Problems
(a) A red die, a blue die, and a yellow die (all six-sided) are rolled. We are interested in
the probability that the number appearing on the blue die is less than that appearing
on the yellow die, which is less than that appearing on the red die. That is, with
B, Y, and R denoting, respectively, the number appearing on the blue, yellow, and
red die, we are interested in P (B < Y < R).
(i) What is the probability that no two of the dice land on the same number?
(ii) Given that no two of the dice land on the same number, what is the conditional
probability that B < Y < R?
(iii) What is P (B < Y < R)?
(b) The following method was proposed to estimate the number of people over the age of
50 who reside in a town of known population 100,000: ”As you walk along the streets,
keep a running count of the percentage of people you encounter who are over 50. Do
this for a few days; then multiply the percentage you obtain by 100,000 to obtain the
estimate.” Comment on this method.
Hint: Let p denote the proportion of people in the town who are over 50. Furthermore, let α1 denote the proportion of time that a person under the age of 50 spends
in the streets, and let α2 be the corresponding value for those over 50. What quantity
does the method suggested estimate? When is the method approximately equal to p?
(c) On rainy days, Joe is late to work with probability .3; on non-rainy days, he is late
with probability 0.1. With probability 0.7, it will rain tomorrow.
(i) Find the probability that Joe is early tomorrow morning.
(ii) Given that Joe was early, what is the conditional probability that it rained?
(d) Suppose than an insurance company classifies people into one of three classes: good
risks, average risks, and bad risks. The company’s records indicate that the probabilities that good-, average-, and bad-risk persons will be involved in an accident over a
1-year span are, respectively, 0.05, 0.15, and 0.3. If 20 percent of the population is a
good risk, 50 percent of the population is an average risk, and 30 percent a bad risk,
what proportion of people have accidents in a fixed year? If policyholder A had no
accidents in 1997, what is the probability that he or she is a good or average risk?
2. Theoretical Exercises
(e) A ball is in any one of n boxes and is in the ith box with probability Pi . If the ball
is in box i, a search of that box will uncover it with probability αi . Show that the
conditional probability that the ball is in box j, given that a search of box i did not
uncover it, is
P
(i) 1−αji Pi if j 6= i
(ii)
(1−αi )Pi
1−αi Pi
if j = i
1